On Bicomplex Representation Methods and Applications of Matrices over Quaternionic Division Algebra

ABSTRACT

In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse concept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion algebra problems can be transformed into complex algebra problems to express and study. These concepts can perfect the theory of [J.L. Wu, A new representation theory and some methods on quaternion division algebra, JP Journal of Algebra, 2009, 14(2): 121-140] and unify the complex algebra and quaternion division algebra.

In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse concept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion algebra problems can be transformed into complex algebra problems to express and study. These concepts can perfect the theory of [J.L. Wu, A new representation theory and some methods on quaternion division algebra, JP Journal of Algebra, 2009, 14(2): 121-140] and unify the complex algebra and quaternion division algebra.

KEYWORDS

Quaternion Determinant, Product of Quaternion Matrix, Inverse of Quaternion Matrix, Similar Quaternion Matrix, Application, Solution

Quaternion Determinant, Product of Quaternion Matrix, Inverse of Quaternion Matrix, Similar Quaternion Matrix, Application, Solution

Cite this paper

nullJ. Wu and P. Zhang, "On Bicomplex Representation Methods and Applications of Matrices over Quaternionic Division Algebra,"*Advances in Pure Mathematics*, Vol. 1 No. 2, 2011, pp. 9-15. doi: 10.4236/apm.2011.12004.

nullJ. Wu and P. Zhang, "On Bicomplex Representation Methods and Applications of Matrices over Quaternionic Division Algebra,"

References

[1] L. X. Chen, “Inverse Matrix and Properties of Double Deter-minant over Quaternion TH Field,” Science in China (Series A), Vol. 34, No. 5, 1991, pp. 25-35.

[2] L. X. Chen, “Generaliza-tion of Cayley-Hamilton Theorem over Quaternion Field,” Chinese Science Bulletin, Vol. 17, No. 6, 1991, pp. 1291-1293.

[3] R, M. Hou, X. Q. Zhao and l. T. Wang, “The Double Determinant of Vandermonde’s Type over Quaternion Field,” Applied Mathematics and Mechanics, Vol. 20, No. 9, 1999, pp. 100-107.

[4] L. P. Huang, “The Determinants of Quateruion Matrices and Their Propoties,” Journal of Mathe-matics Study, Vol. 2, 1995, pp. 1-13.

[5] J. L. Wu, L. M. Zou, X. P. Chen and S. J. Li, “The Estimation of Eigenvalues of Sum, Difference, and Tensor Product of Matrices over Quater-nion Division Algebra,” Linear Algebra and its Applications, Vol. 428, 2008, pp. 3023-3033. doi:10.1016/j.laa.2008.02.008

[6] T. S. Li, “Properties of Double Determinant over Quaternion Field,” Journal of Cen-tral China Normal University, Vol. 1, 1995, 3-7. doi:10.1007/BF02652076

[7] B. X. Tu, “Dieudonne Deter-minants of Matrices over a Division Ring,” Journal of Fudan university, 1990A, Vol. 1, pp. 131-138.

[8] B. X. Tu, “Weak Direct Products and Weak Circular Product of Matrices over the Real Quaternion Division Ring,” Journal of Fudan Univer-sity, Vol. 3, 1991, p. 337.

[9] J. L. Wu, “Distribution and Estimation for Eigenvalues of Real Quaternion Matrices,” Computers and Mathematics with Applications, Vol. 55, 2008, pp. 1998-2004. doi:10.1016/j.camwa.2007.07.013

[10] B. J. Xie, “Theorem and Application of Determinants Spread out of Self-Conjugated Matrix,” Acta Mathematica Sinica, Vol. 5, 1980, pp. 678-683.

[11] Q. C. Zhang, “Properties of Double Determinant over the Quaternion Field and Its Applications,” Acta Mathe-matica Sinica, Vol. 38, No. 2, 1995, pp. 253-259.

[12] W. J. Zhuang, “Inequalities of Eigenvalues and Singular Values for Quaternion Matrices,” Advances in Mathematics, Vol. 4, 1988, pp. 403-406.

[13] W. Boehm, “On Cubics: A Survey, Com-puter Graphics and Image Processing,” Vol. 19, 1982, pp. 201-226. doi:10.1016/0146-664X(82)90009-0

[14] G. Farin, “Curves and Surfaces for Computer Aided Geometric Design,” Aca-demic Press, Inc., San Diego CA, 1990.

[15] K. Shoemake, “Animating Rotation with Quaternion Calculus,” ACM SIG-GRAPH, 1987, Course Notes, Computer Animation: 3–D Mo-tion, Specification, and Control.

[16] Q. G. Wang, “Quater-nion Transformation and Its Application to the Displacement Analysis of Spatial Mechanisms, Acta Mathematica Sinica, Vol. 15, No. 1, 1983, pp. 54-61.

[17] G. S. Zhang, “Commutativity of Composition for Finite Rotation of a Rigid Body,” Acta Mechanica Sinica, Vol. 4, 1982.

[18] E. T. Browne, “The Characteristic Roots of a Matrix,” Bulletin of the American Mathematical Society, Vol. 36, 1930, pp. 705-710. doi:10.1090/S0002-9904-1930-05041-7

[19] J. L. Wu and Y. Wang, “A New Representation Theory and Some Methods on Quaternion Division Algebra,” Journal of Algebra, Vol. 14, No. 2, 2009, pp. 121-140.

[20] Q. W. Wang, “The General Solu-tion to a System of Real Quaternion Matrix Equation,” Com-puter and Mathematics with Applications, Vol. 49, 2005, pp. 665-675. doi:10.1016/j.camwa.2004.12.002

[21] G. B. Price, “An In-troduction to Multicomplex Spaces and Functions,” Marcel Dekker, New York, 1991.

[22] D. Rochon, “A Bicomplex Riemann Zeta Function,” Tokyo Journal of Mathematics, Vol. 27, No. 2, 2004, pp. 357-369.

[23] S. P. Goyal and G. Ritu, “The Bicomplex Hurwitz Zeta function,” The South East Asian Journal of Mathematics and Mathematical Sciences, 2006.

[24] S. P. Goyal, T. Mathur and G. Ritu, “Bicomplex Gamma and Beta Function,” Journal of Raj. Academy Physical Sciences, Vol. 5, No. 1, 2006, pp. 131-142.

[25] J. N. Fan, “Determinants and Multiplicative Functionals on Quaternion Matrices,” Linear Algebra and Its Applications, Vol. 369, 2003, pp. 193-201. doi:10.1016/S0024-3795(02)00722-X

[26] Q. W. Wang, “A System of Four Matrix Equations over Von Neumann Regular Rings and It Applications,” Acta Mathematica Sinica, Vol. 21, 2005, pp. 323-334. doi:10.1007/s10114-004-0493-1

[27] Q. W. Wang, “A System of Matrix Equation and a Linear Matrix Equation over Arbi-trary Regular Rings with Identity,” Applied Linear Algebra, Vol. 384, 2004, pp. 43-54. doi:10.1016/j.laa.2003.12.039

[28] W. J. Zhuang, “The Guide of Matrix Theory over Quaternion Field,” Science Press, Bei-jing, 2006, pp. 1-50.

[29] W. L. LI, “Quaternion Matrices,” “National Defense Science and Technology University,” Vol. 6, 2002, pp. 73-74.

[30] R. X. Jiang, “Linear Algebra,” People’s Educational Press, China, 1979, pp. 41-42.

[1] L. X. Chen, “Inverse Matrix and Properties of Double Deter-minant over Quaternion TH Field,” Science in China (Series A), Vol. 34, No. 5, 1991, pp. 25-35.

[2] L. X. Chen, “Generaliza-tion of Cayley-Hamilton Theorem over Quaternion Field,” Chinese Science Bulletin, Vol. 17, No. 6, 1991, pp. 1291-1293.

[3] R, M. Hou, X. Q. Zhao and l. T. Wang, “The Double Determinant of Vandermonde’s Type over Quaternion Field,” Applied Mathematics and Mechanics, Vol. 20, No. 9, 1999, pp. 100-107.

[4] L. P. Huang, “The Determinants of Quateruion Matrices and Their Propoties,” Journal of Mathe-matics Study, Vol. 2, 1995, pp. 1-13.

[5] J. L. Wu, L. M. Zou, X. P. Chen and S. J. Li, “The Estimation of Eigenvalues of Sum, Difference, and Tensor Product of Matrices over Quater-nion Division Algebra,” Linear Algebra and its Applications, Vol. 428, 2008, pp. 3023-3033. doi:10.1016/j.laa.2008.02.008

[6] T. S. Li, “Properties of Double Determinant over Quaternion Field,” Journal of Cen-tral China Normal University, Vol. 1, 1995, 3-7. doi:10.1007/BF02652076

[7] B. X. Tu, “Dieudonne Deter-minants of Matrices over a Division Ring,” Journal of Fudan university, 1990A, Vol. 1, pp. 131-138.

[8] B. X. Tu, “Weak Direct Products and Weak Circular Product of Matrices over the Real Quaternion Division Ring,” Journal of Fudan Univer-sity, Vol. 3, 1991, p. 337.

[9] J. L. Wu, “Distribution and Estimation for Eigenvalues of Real Quaternion Matrices,” Computers and Mathematics with Applications, Vol. 55, 2008, pp. 1998-2004. doi:10.1016/j.camwa.2007.07.013

[10] B. J. Xie, “Theorem and Application of Determinants Spread out of Self-Conjugated Matrix,” Acta Mathematica Sinica, Vol. 5, 1980, pp. 678-683.

[11] Q. C. Zhang, “Properties of Double Determinant over the Quaternion Field and Its Applications,” Acta Mathe-matica Sinica, Vol. 38, No. 2, 1995, pp. 253-259.

[12] W. J. Zhuang, “Inequalities of Eigenvalues and Singular Values for Quaternion Matrices,” Advances in Mathematics, Vol. 4, 1988, pp. 403-406.

[13] W. Boehm, “On Cubics: A Survey, Com-puter Graphics and Image Processing,” Vol. 19, 1982, pp. 201-226. doi:10.1016/0146-664X(82)90009-0

[14] G. Farin, “Curves and Surfaces for Computer Aided Geometric Design,” Aca-demic Press, Inc., San Diego CA, 1990.

[15] K. Shoemake, “Animating Rotation with Quaternion Calculus,” ACM SIG-GRAPH, 1987, Course Notes, Computer Animation: 3–D Mo-tion, Specification, and Control.

[16] Q. G. Wang, “Quater-nion Transformation and Its Application to the Displacement Analysis of Spatial Mechanisms, Acta Mathematica Sinica, Vol. 15, No. 1, 1983, pp. 54-61.

[17] G. S. Zhang, “Commutativity of Composition for Finite Rotation of a Rigid Body,” Acta Mechanica Sinica, Vol. 4, 1982.

[18] E. T. Browne, “The Characteristic Roots of a Matrix,” Bulletin of the American Mathematical Society, Vol. 36, 1930, pp. 705-710. doi:10.1090/S0002-9904-1930-05041-7

[19] J. L. Wu and Y. Wang, “A New Representation Theory and Some Methods on Quaternion Division Algebra,” Journal of Algebra, Vol. 14, No. 2, 2009, pp. 121-140.

[20] Q. W. Wang, “The General Solu-tion to a System of Real Quaternion Matrix Equation,” Com-puter and Mathematics with Applications, Vol. 49, 2005, pp. 665-675. doi:10.1016/j.camwa.2004.12.002

[21] G. B. Price, “An In-troduction to Multicomplex Spaces and Functions,” Marcel Dekker, New York, 1991.

[22] D. Rochon, “A Bicomplex Riemann Zeta Function,” Tokyo Journal of Mathematics, Vol. 27, No. 2, 2004, pp. 357-369.

[23] S. P. Goyal and G. Ritu, “The Bicomplex Hurwitz Zeta function,” The South East Asian Journal of Mathematics and Mathematical Sciences, 2006.

[24] S. P. Goyal, T. Mathur and G. Ritu, “Bicomplex Gamma and Beta Function,” Journal of Raj. Academy Physical Sciences, Vol. 5, No. 1, 2006, pp. 131-142.

[25] J. N. Fan, “Determinants and Multiplicative Functionals on Quaternion Matrices,” Linear Algebra and Its Applications, Vol. 369, 2003, pp. 193-201. doi:10.1016/S0024-3795(02)00722-X

[26] Q. W. Wang, “A System of Four Matrix Equations over Von Neumann Regular Rings and It Applications,” Acta Mathematica Sinica, Vol. 21, 2005, pp. 323-334. doi:10.1007/s10114-004-0493-1

[27] Q. W. Wang, “A System of Matrix Equation and a Linear Matrix Equation over Arbi-trary Regular Rings with Identity,” Applied Linear Algebra, Vol. 384, 2004, pp. 43-54. doi:10.1016/j.laa.2003.12.039

[28] W. J. Zhuang, “The Guide of Matrix Theory over Quaternion Field,” Science Press, Bei-jing, 2006, pp. 1-50.

[29] W. L. LI, “Quaternion Matrices,” “National Defense Science and Technology University,” Vol. 6, 2002, pp. 73-74.

[30] R. X. Jiang, “Linear Algebra,” People’s Educational Press, China, 1979, pp. 41-42.